Observables in loop quantum gravity with a cosmological constant

Abstract

An open issue in loop quantum gravity (LQG) is the introduction of a non-vanishing cosmological constant Λ. In 3d, Chern-Simons theory provides some guiding lines: Λ appears in the quantum deformation of the gauge group. The Turaev-Viro model, which is an example of spin foam model is also defined in terms of a quantum group. By extension, it is believed that in 4d, a quantum group structure could encode the presence of Λ = 0. In this article, we introduce by hand the quantum group Uq(su(2)) into the LQG framework, that is we deal with Uq(su(2))-spin networks. We explore some of the consequences, focusing in particular on the structure of the observables. Our fundamental tools are tensor operators for Uq(su(2)). We review their properties and give an explicit realization of the spinorial and vectorial ones. We construct the generalization of the U(N ) formalism in this deformed case, which is given by the quantum group Uq(u(n)). We are then able to build geometrical observables, such as the length, area or angle operators ... We show that these operators characterize a quantum discrete hyperbolic geometry in the 3d LQG case. Our results confirm that the use of quantum group in LQG can be a tool to introduce a non-zero cosmological constant into the theory. ContentsIntroduction 2 I. U q (su(2)) in a nutshell 4 A. Definition of U q (su(2)) 4 B. q-harmonic oscillators and the Schwinger-Jordan trick 7II. Tensor operators for U q (su(2)) 7 A. Definition and Wigner-Eckart theorem 7 B. Product of tensor operators: scalar product, vector product and triple product 8 1. Scalar product 9 2. Vector product 10 C. Tensor products of tensor operators 10 III. Realization of tensor operators of rank 1/2 and 1 for U q (su(2)) 11 A. Rank 1/2 tensor operators 11 B. Rank 1 tensor operators 13IV. Observables for the intertwiner space 16 A. General construction and properties of intertwiner observables 16 B. U q (u(n)) formalism for LQG defined over U q (su (2)) 18 [6,7]. Indeed, in a 3d space-time, one can rewrite General Relativity with a (possibly zero) cosmological constant as a Chern-Simons gauge theory 1 . The general phase space structure of the theory for any metric signature and sign of Λ can be treated in a nice unified way [8], using Poisson-Lie groups [9], the classical counterparts of quantum groups. The quantization procedure leads explicitly to a quantum group structure. The full construction, from phase space to quantum group is usually called combinatorial quantization [5][6][7]. We can also quantize 3d gravity using the spinfoam approach. In this approach, 3d gravity is formulated as a BF theory. When Λ = 0, this is the well-known Ponzano-Regge model (both Euclidian or Lorentzian), based on the irreducible unitary representations of the relevant gauge group. When Λ = 0, the quantum group structure is introduced by hand. The Ponzano-Regge model is deformed, using irreducible unitary representations of the relevant quantum deformation of the gauge group. This is then called the Turaev-Viro model [10]. The argument co...

(3 + 1)-dimensional topological phases and self-dual quantum geometries encoded on Heegaard surfaces

“…With a cosmological constant one would, however, expect a vacuum describing a homogeneously curved geometry. There are a number of approaches to incorporate homogeneously curved geometry into the kinematical set-up of (loop) quantum gravity [18][19][20][21]. An important aspect, making such a construction very attractive [22], is that one expects the Hilbert space associated to a fixed triangulation 1 to be finite dimensional.…”

“…In addition to Wilson loop operators one can consider grasping operators [18], which usually implement the fluxes, and can be straightforwardly evaluated using (3.3), (3.4). How are these operators geometrically interpreted [21,85]? Do they provide an alternative quantization of (possibly non-exponentiated) flux operators?…”

The closure constraint for the hyperbolic tetrahedron as a Bianchi identity

“…In this papier, we provided a first step towards validating the geometrical interpretation of the spin network states of q-deformed loop quantum gravity in 3+1-dimensions [27][28][29][30] as discrete hyperbolic geometry. Indeed we showed that the SB(2, C) Gauss law or closure constraints imposed at the spin network vertices are related to hyperbolic tetrahedra, mimicking the correspondance given by the Minkowski theorem in the flat case between R 3 closure constraints and (convex) polyhedra.…”

“…We discuss the splitting of the natural SL(2, C) closure with respect to the Iwasawa decomposition and study the possibility of the reconstruction of the original hyperbolic tetrahedron. Finally, we check that, in the framework of the q-deformed phase for loop quantum gravity [27][28][29], the defined closure relation generates 3d rotations of the normals and of the hyperbolic tetrahedron as expected. …”

“…For instance, for the covariant approach, the Ponzano-Regge model is deformed into the Turaev-Viro model based on the SU q (2). At the canonical level, several directions are pursued, either by putting the quantum group structure directly at the kinematical level [27][28][29][30] or by trying to retrieve it through the dynamics [31]. In this new canonical development, we expect a kind of quantum deformed twisted geometry to appear, and indeed, the variables associated to the classical limit of the quantum deformation corresponds to the construction of hyperbolic polygons (at least in the 3-valent case) [28] and the spin networks allow the gluing of such polygons to construct a whole space.…”